Distance is the cheapest and most immediately available tool in radiation protection. Double your distance from a point source and the dose rate drops to one quarter. Triple the distance and it drops to one ninth. This relationship, the inverse square law, governs every boundary rope placement, every radiography shot plan, and every emergency response decision involving a discrete source.
The inverse square law is not an approximation or a rule of thumb. It is a direct consequence of geometry: radiation from a point source spreads uniformly over the surface of an expanding sphere, and the area of that sphere increases with the square of the radius. This guide covers the math, the assumptions that must hold for the law to apply, common field calculations, and the regulatory boundaries defined in 10 CFR 20 that rely on dose rate at specific distances.
The Geometry Behind the Law
A point source emits radiation uniformly in all directions. At a distance d from the source, that radiation is spread over the surface of a sphere with area 4πd². At twice the distance, the sphere has four times the surface area. The same total number of photons (or particles) passes through each sphere per unit time, but the flux per unit area, which is what your dosimeter measures, decreases by the factor (d&sub1;/d&sub2;)².
The formula is: D&sub2; = D&sub1; × (d&sub1;/d&sub2;)², where D&sub1; is the known dose rate at distance d&sub1;, and D&sub2; is the dose rate at the new distance d&sub2;. This can also be written as D&sub1; × d&sub1;² = D&sub2; × d&sub2;², which says the product of dose rate and distance squared is constant for a given source.
The law applies strictly to point sources in a vacuum with no scattering medium. In practice, it works well for sealed sources where the source-to-detector distance is at least five times the largest source dimension. A 3 mm diameter Ir-192 source at 15 mm or more behaves as a point source to within a few percent. At shorter distances, the geometry correction for an extended source becomes significant.
Inverse Square Law: D&sub2; = D&sub1; × (d&sub1; / d&sub2;)²
Equivalently: D&sub1; × d&sub1;² = D&sub2; × d&sub2;² (the product of dose rate and distance squared is constant for a given point source).
Radiation Distance Calculator
Calculate dose rate at any distance from a radiation source using the inverse square law. Returns dose falloff table, 2 mR/hr boundary distance per 10 CFR 20.1301, and High Radiation Area boundary.
When the Inverse Square Law Applies (and When It Does Not)
The inverse square law holds when three conditions are met: the source behaves as a point, the medium between source and detector does not significantly attenuate or scatter the beam, and there are no nearby surfaces reflecting radiation back toward the detector.
Extended sources: A spent fuel assembly, a contaminated pipe wall, or a large-area surface source does not follow the inverse square law at close range. The dose rate from a line source falls off as 1/d (not 1/d²) at distances much smaller than the line length, and a large plane source produces a nearly constant dose rate at very close range. Once you are far enough away that the entire source subtends a small solid angle, it starts behaving like a point source again.
Air attenuation: For gamma energies above about 200 keV and distances under 100 meters, air attenuation is negligible and the law holds. For low-energy photons (Am-241 at 60 keV, for example) or for very long distances, air attenuation reduces the dose rate below what the inverse square law alone would predict. At industrial radiography distances (typically under 50 meters with Ir-192 or Co-60), air attenuation is not a significant factor.
Scatter and buildup: In a shielded room, radiation scatters off walls, floors, and ceilings and adds to the direct beam at the detector location. The inverse square law predicts only the direct (unscattered) component. In a small concrete vault, scattered radiation can add 10 to 30 percent above the inverse-square prediction at certain locations. For open-air radiography, scatter from the ground is usually small but not zero.
The inverse square law applies to point sources. For contaminated surfaces, line sources, or large distributed sources at close range, the falloff is slower than 1/d². Survey the actual field with instruments rather than relying on calculation alone.
Common Field Calculations
The most frequent field calculation is: "I measured X mR/hr at distance d&sub1;. What is the dose rate at distance d&sub2;?" This arises constantly during radiography, source exchanges, leak testing, and emergency response.
Example 1: A radiographer measures 200 mR/hr at 2 meters from an Ir-192 source. What is the dose rate at 10 meters? D&sub2; = 200 × (2/10)² = 200 × 0.04 = 8 mR/hr. This is well above the 2 mR/hr unrestricted area limit, so the boundary must be farther out.
Example 2: At what distance does the dose rate drop to 2 mR/hr? Rearrange the formula: d&sub2; = d&sub1; × sqrt(D&sub1;/D&sub2;) = 2 × sqrt(200/2) = 2 × 10 = 20 meters. The boundary rope must be at least 20 meters from the source in every direction.
Example 3: Two survey readings at different distances can verify that the field is behaving as an inverse-square field. If you measure 50 mR/hr at 3 meters and 12.5 mR/hr at 6 meters, the ratio is 50/12.5 = 4.0, and (6/3)² = 4.0. The readings are consistent. If the ratio does not match, investigate: there may be a second source, scatter contribution, or instrument error.
Always carry the calculation through with units and sanity-check the result. If your boundary distance comes out to 200 meters for a 10 Ci Ir-192 source, something is off. Typical unrestricted area boundaries for a 100 Ci Ir-192 source in open-air radiography are on the order of 40 to 70 meters depending on collimation.
To find the distance where dose rate drops to a target value: d&sub2; = d&sub1; × sqrt(D&sub1; / D&sub2;). This is the boundary distance formula used for every radiography shot plan and restricted area posting.
Radiation Distance Calculator
Calculate dose rate at any distance from a radiation source using the inverse square law. Returns dose falloff table, 2 mR/hr boundary distance per 10 CFR 20.1301, and High Radiation Area boundary.
Regulatory Boundaries Defined by Dose Rate
The NRC defines radiation area classifications in 10 CFR 20.1003 based on dose rate. These classifications trigger specific posting and access control requirements under 10 CFR 20.1902:
- Radiation Area: An area where an individual could receive a dose equivalent in excess of 5 mrem in one hour at 30 cm from the source or from any surface that the radiation penetrates. Posted with "CAUTION, RADIATION AREA" sign.
- High Radiation Area: An area where an individual could receive a dose equivalent in excess of 100 mrem in one hour at 30 cm. Posted with "CAUTION, HIGH RADIATION AREA" or "DANGER, HIGH RADIATION AREA" sign. Requires access control per 10 CFR 20.1601 or 10 CFR 20.1602.
- Very High Radiation Area: An area where an individual could receive an absorbed dose in excess of 500 rads in one hour at 1 meter from the source or from any surface that the radiation penetrates. Requires additional controls per 10 CFR 20.1602.
The inverse square law is the calculation method used to determine at what distance these thresholds are crossed. Given a known dose rate at the source (or at a measured reference distance), calculate the distance at which the dose rate equals 5 mR/hr, 100 mR/hr, or 500 R/hr. Everything inside that distance requires the corresponding posting and controls.
Note that the measurement point for Radiation Area and High Radiation Area is at 30 cm from the source or surface, not at contact. This distinction matters for small sealed sources in shielded containers where the dose rate at contact may be high but drops rapidly with distance.
10 CFR 20.1003 Thresholds: Radiation Area: >5 mrem/hr at 30 cm. High Radiation Area: >100 mrem/hr at 30 cm. Very High Radiation Area: >500 rad/hr at 1 m. These classifications drive posting and access control requirements.
Industrial Radiography: Applying Distance in the Field
Industrial radiography under 10 CFR 34 is the most common field application of the inverse square law. Every radiography shot requires establishing a restricted area boundary where the dose rate does not exceed 2 mR/hr (the public dose limit equivalent). The radiographer calculates this distance from the source activity, the specific gamma ray constant, any collimation factor, and the inverse square law.
In practice, radiographers carry a reference table showing boundary distances for their sources at various activities. For Ir-192, a 100 Ci source has a dose rate of approximately 480 R/hr at 1 foot (based on the gamma constant of 4.69 R/hr per Ci at 1 meter, converted to 1 foot). To find the 2 mR/hr boundary, solve for d where 480,000 mR/hr × (1 ft)² = 2 mR/hr × d². This gives d = sqrt(240,000) = 490 feet, or about 150 meters.
This is why pipeline radiography in open areas requires large exclusion zones, and why radiography in fabrication shops is done in shielded enclosures (vaults) rather than relying on distance alone. A 20-foot-thick concrete vault provides enough shielding that the dose rate outside the walls is well below 2 mR/hr regardless of source activity, eliminating the need for large standoff distances.
Collimators reduce the boundary distance in directions where they block the beam. A collimator that attenuates the beam by a factor of 100 in the backward direction reduces the boundary in that direction by a factor of 10 (sqrt of 100). This is why directional exposure devices are preferred for field radiography: they reduce the exclusion zone on the side away from the shot.
Never rely on calculation alone for setting radiography boundaries. Always survey with a calibrated instrument to confirm that the dose rate at the boundary is at or below 2 mR/hr. Scatter, ground reflection, and collimator leakage can all increase the actual field above the calculated value.
Radiation Distance Calculator
Calculate dose rate at any distance from a radiation source using the inverse square law. Returns dose falloff table, 2 mR/hr boundary distance per 10 CFR 20.1301, and High Radiation Area boundary.
Multiple Sources and Combined Fields
When more than one source contributes to the dose rate at a location, the dose rates add. Each source follows the inverse square law independently, and the total dose rate at any point is the sum of the individual contributions. This is straightforward in principle but requires careful bookkeeping in practice.
A common scenario is a radiography vault with a stored source and an active source. Even though the stored source is in its shielded container, it contributes some dose rate at the operator's position. If the stored source contributes 0.5 mR/hr and the active source (through the vault wall) contributes 1.2 mR/hr, the operator's total dose rate is 1.7 mR/hr. Both contributions must be considered when assessing the operator's daily exposure.
For nuclear gauge users who store multiple gauges in a single cabinet or vehicle, the total dose rate at the surface of the cabinet is the sum of the dose rates from all gauges. Two Cs-137 moisture-density gauges stored side by side produce roughly twice the dose rate of one gauge at the same measurement point. The storage location and vehicle labeling requirements depend on this combined field, not the individual gauge contributions.
To find the location where the combined field from multiple sources equals a regulatory threshold, you generally need to solve iteratively or use a calculator. The boundary is no longer a simple circle around a single source; it becomes an irregular contour that bulges toward the midpoint between sources where the fields overlap most.
Practical Tips for Distance-Based Protection
Step back before you think. If you find yourself in an unexpectedly high field (your rate alarm sounds, your APD chirps faster than expected), the correct first action is to increase distance immediately. Step back 10 feet and then assess. Do not stand in the field trying to figure out what went wrong.
Use the factor-of-four rule. Doubling distance reduces dose rate by a factor of four. This is easier to remember and apply than the full formula in a field situation. If you are at 5 mR/hr and need to be below 2 mR/hr, one doubling of distance gets you to 1.25 mR/hr. That is usually enough.
Measure, do not assume. The inverse square law tells you what to expect. Your survey instrument tells you what is actually there. Any discrepancy between calculation and measurement means something in your assumptions is wrong: a second source you did not account for, scatter from a nearby wall, a collimator leak, or an instrument calibration problem. Investigate the discrepancy.
Account for time in the field. A low dose rate does not mean a low dose if you spend a long time at that distance. Standing at 0.5 mR/hr for an 8-hour shift gives you 4 mR, which is 0.8% of the annual occupational limit. For a single shift that is trivial, but for a full year of daily exposure at that rate, you would accumulate over 1 rem. Distance planning must be combined with time planning for an effective ALARA program.
The factor-of-four rule: doubling your distance from a point source cuts the dose rate to one quarter. This is the single most useful mental shortcut in radiation protection fieldwork.